Question 1186926
To determine the most cost-effective mix of Rabbit-Gro and Lucky-Rabbit food that meets the rabbits' nutritional needs, we can use a system of inequalities and solve for the optimal amounts of each food.

**Let:**

*   x = ounces of Rabbit-Gro
*   y = ounces of Lucky-Rabbit

**Objective:** Minimize cost: C = 0.20x + 0.30y

**Constraints:**

*   **Fat:** 8x + 12y ≥ 24
*   **Carbohydrates:** 12x + 12y ≥ 36
*   **Protein:** 2x + y ≥ 4
*   **Total Food:** x + y ≤ 5
*   **Non-negativity:** x ≥ 0, y ≥ 0

**Solving the Problem:**

One way to solve this is graphically. Plot each inequality on a graph with x and y axes. The feasible region is where all inequalities are satisfied. The optimal solution will be at one of the vertices (corners) of this region.

Alternatively, you can use linear programming techniques (like the simplex method or software tools) to find the optimal solution.

**Graphical Method (Sketch):**

1.  **Plot the lines:** Treat each inequality as an equation and plot the lines on a graph.
2.  **Shade the regions:** Shade the appropriate side of each line based on the inequality sign.
3.  **Identify the feasible region:** The feasible region is where all shaded areas overlap.
4.  **Find the vertices:** Determine the coordinates of the vertices of the feasible region.
5.  **Evaluate the objective function:** Plug the x and y coordinates of each vertex into the cost equation (C = 0.20x + 0.30y).
6.  **Optimal solution:** The vertex that yields the lowest cost is the optimal solution.

**Linear Programming:**

Linear programming is a more efficient way to solve this type of optimization problem, especially if there are many variables or constraints.

**Expected Outcome:**

By solving the system of inequalities and minimizing the cost function, you'll find the optimal number of ounces of Rabbit-Gro and Lucky-Rabbit to feed the rabbits while meeting their nutritional requirements and staying within the 5-ounce limit.

*The solution will likely involve a combination of both foods, but it's possible that one food alone might be the most cost-effective option.*